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MIXING CONDITIONS AND RETURN TIMES
ON MARKOV TOWERS
by
Yiannis Psiloyenis
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
August 2008
Copyright 2008 Yiannis Psiloyenis

This dissertation discusses mixing properties derived on non-uniformly hyperbolic dynamical systems which admit a Markov-Tower structure. For systems with exponential and polynomial decay of the tails of the return map we derive alpha-mixing conditions at exponential and polynomial rates, respectively. The motivation is to use these mixing conditions, on eligible systems, to approximate the law of the hitting and return times to a set of small measure.; In the first chapter we set up the problem and derive mixing conditions. In the next chapter we give a brief discussion on Stein method and in chapter three we study the hitting and multiple-return times for α-mixing dynamical systems in general. Under the given rates of mixing we use the Stein method to show that the return time of order k to a cylinder A can be approximated by a simple distribution for which sharp error bounds, independent of the order k, are obtained as well. Additionally, we conclude that the distribution of hitting times, suitably rescaled, can be approximated by an exponential distribution with mean 1.; As an application we show that the findings for the return and hitting times can be applied, through the construction of aMarkov Tower, to the Gaspard-Wang map which is broadly used in Physics.

MIXING CONDITIONS AND RETURN TIMES
ON MARKOV TOWERS
by
Yiannis Psiloyenis
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
August 2008
Copyright 2008 Yiannis Psiloyenis